The numerical framework and simulation setup presented herein have previously been carefully validated in the context of elongated bubbles in circular capillary channels. For this validation, we refer to the work of [1]}, in which the code was benchmarked against the well-known correlation of [2]} for steady-state liquid film thickness in conditions of non-negligible inertia. For \(Ca \sim \)  O(\(10^{-2}\) ) and \(Re \sim \)  O(\(10^{2}\) ), deviations of up to \(10\%\)  were reported, which are within the uncertainty of the correlation. A supplementary validation for square capillary channels is conducted using the numerical data and conditions described in [3]} for surfactant-free scenarios under non-negligible inertia. Satisfactory quantitative agreement between our results and those of [3]} is achieved in terms of bubble-to-liquid velocity ratio, \(U_{b}/U_{l}\) , and gas area fraction, \(\epsilon \)  (see figure REF (a)-(b)), with maximum deviations of 2.7% and 3.1%, respectively. We underline that the cases that do not have a direct counterpart from the above mentioned reference (i.e., \(Ca = 0.0089\) , \(Ca = 0.0377\) , and \(Ca = 0.068\)  for \(Re = 443\) ) adequately follow the qualitative trend of increasing \(U_{b}/U_{l}\)  and decreasing \(\epsilon \)  with \(Ca\)  and \(Re\) .
<FIGURE>